Rounding Error Propagation: Bias and Uncertainty (original) (raw)
2024, ForsChem Research Reports
Any rounding operation of a value causes loss of information, and thus, introduces error. Two types of error are involved: Systematic error (bias) and random error (uncertainty). Uncertainty is always introduced for any type of rounding employed. Bias is directly introduced only when lower ("floor") and upper ("ceiling") types of rounding are used. Central rounding is in principle unbiased, but bias may emerge in the case of nonlinear operations. The purpose of this report is discussing the propagation of both types of rounding error when rounded values are used in common mathematical operations. The basic mathematical operations considered are addition/subtraction, product, and natural powers. These operations can be used to evaluate the propagation of error in power series, which then are used to describe error propagation for any arbitrary nonlinear function. Even when power series approximations can be obtained for any arbitrary reference value, it is highly recommended using the corresponding rounded value as reference. The error propagation expressions obtained are implemented in R language to facilitate the calculations. A couple of examples are included to illustrate the evaluation of error propagation. These examples also show that truncating the power series after the linear term already provides a good estimation of error propagation (using the rounded value as reference point for the power series expansion).
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